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The relationship between mass transport to channel and rotating disk electrodes
287
J. Electroanal. Chem., 245 (1988) 287-298 Elsevier Sequoia S.A., Lausanne - Printed
in The Netherlands
THE RELATIONSHIP BETWEEN MASS TRANSPORT AND ROTATING DISK ELECTRODES
PATRICK
R. UNWIN
and RICHARD
Physical Chemistry Laboratory, (Received
TO CHANNEL
G. COMPTON
South Parks Road, Oxford OXI 3Q.Z (Great Britain)
29th June 1987; in revised form 29th September
1987)
ABSTRACT A simple equation is presented which relates the diffusion layer thickness at a rotating disk electrode to the average diffusion layer thickness at a channel (or tubular) electrode, under the assumption that the hydrodynamic flow in both cases is laminar. It is demonstrated that, through the equation, the parameters characterising various possible electrode reaction mechanisms (ECE, DISPl, EC’, CE,.. .) under transport-limited conditions at one of the two electrodes, are readily deduced once the corresponding mass transport problem has been rigorously solved at the other electrode. In addition, the relationship is shown to be equally successful when considering the chronoamperometric response at the two electrodes to a potential step, from a region in which no current flows to one corresponding, under steady-state conditions, to the passage of the transport-limited current, and also for the description of the general form of the current-potential curves for the two electrode geometries. The circumstances under which the transformation may be applied are critically assessed.
INTRODUCTION
The current intense activity in the characterisation of electrode reaction mechanisms has led to the theoretical description of a wide range of mechanisms at both rotating disc (RDE) and channel (CE) electrodes. The former constitutes probably the most commonly employed type of (solid) hydrodynamic electrode, whereas the advantages of the latter have been described recently [1,2]. Superficially the problem of mass transport to the two electrodes appears quite different, as can be seen by considering the relevant convective-diffusion equations for both cases. For the transport of a kinetically stable species of concentration, c, to a RDE we have [3],
ac a% ;i;=Dax’-““ax
ac
u, c - ~.O~~W~‘*Y-“~X~(Xe 0)
(1)
where x is the distance normal to, and starting from the surface of the electrode (see 0022-0728/88/$03.50
0 1988 Elsevier Sequoia
S.A.
288
(a)
2h
Fig. 1. Coordinate system for (a) the channel electrode and (b) the rotating disc electrode.
Fig. l), Y is the kinematic viscosity of the solution, o (Hz) is the disc rotation speed, and D is the diffusion coefficient of the species concerned. In contrast the corresponding CE equation is [4] a~ a2c ~=Dj--y-yjg
ac
ux=“o(l(y2)
(2)
where x is distance along the channel, y is the distance normal to the electrode surface and h is the half-height of the channel (Fig. 1). In writing eqn. (2) we have assumed that diffusion in directions other than normal to the electrode surface may be neglected. This approximation has been thoroughly explored and shown to be entirely satisfactory for practical electrodes [5,6]. In the treatment of CE problems it is usual to invoke the so-called Ltv&que approximation, originally introduced in respect of the corresponding heat transfer problem [7], u,-2u,y/h
(foryc0)
(3)
Comparison of the form at eqns. (1) and (3), which are valid for values of x close to the electrode surface, shows that the velocity profile is parabolic in x at the RDE, but linear at the CE. A further comparison of eqns. (1) and (2) indicates that the concentration profiles at the RDE resulting from the solution of the convective-diffusion equation will depend on x only - the RDE being “uniformly accessible” - but those at the CE will depend on both x and y. However, a further simplification in respect of
289
eqn. (2) has very recently been adopted by Dutt and Sir@ [8]. This involves replacing the concentration gradient in the x-direction by “its average value” [8]. Mathematically this means that for the one-electron electrode reaction, Red%Ox+ewe can write a[Red]/ax
= ([Red] - [Red]“)/l
(4)
and a[ox]/ax
= [ox]/z
(5)
where I is the electrode length and [Red] O is the bulk concentration of the electroactive species ([Ox]” is assumed to be zero). These approximations reduce eqn. (2) to an ordinary differential equation (in JJ). It has been shown previously [9] that theoretical results generated in this way are remarkably successful - when compared with full rigorous theory - for a wide range of problems including both time-dependent phenomena (such as chronoamperometry [9], linear sweep and cyclic voltammetry [8,10-121) and also complex steady-state problems (such as the current-flow rate behaviour of ECE and DISPl processes [9]). In other words, the question of mass transport to the CE can be treated satisfactorily as a one-dimensional problem within an “average” diffusion layer generated by the Singh and Dutt approximation - at least where the problem involves either stable species or species decaying by first-order kinetics [9]. Historically the RDE was the first solid hydrodynamic electrode to be widely adopted (see eg. ref. 13). Only relatively recently have the advantages of the CE been appreciated [1,2] - early workers probably being deterred because of the then wrongly perceived difficulties of handling mass transport problems at non-uniformly accessible electrodes [1,2]. Consequently, the literature describing the parameters (effective number of electrons transferred under limiting current conditions, half-wave potential,. . . ) which characterise the various possible mechanisms (ECE, DISP, EC’, CE, . . . ) o f an electrode reaction is far richer for the RDE than for the CE, although some theoretical treatments for the latter electrode have recently become available [14-171 - generally built on the LCv&que approximation (see above). Such parameters are normally reported in the form of “working curves” in which the dependence of the parameter of interest on an appropriate normalised variable (which depends typically on a rate constant, w or ua and D) is illustrated graphically and this diagram is then used for the analysis of experimental data. Since the publication of various “working curves” for the CE we have been impressed with the apparent similarity in shape of the corresponding curves for the two electrodes, although this is perhaps not unexpected in the light of the Singh and Dutt Approximation. Consequently, we have investigated whether a simple transformation exists by which the behaviour of a particular process at one electrode type may be deduced once the corresponding problem has been solved at the other electrode. This paper suggests the appropriate transformation in terms of a
290
simple direct relationship between the diffusion layer thickness at the two electrodes - and demonstrates the successful application of the transformation in a diversity of problems. The conditions under which the transformation may be applied are carefully assessed. THE TRANSFORMATION
AND ITS APPLICATIONS
We suggest that, to a reasonable approximation, the diffusion layer thickness, S,, at the RDE can be related to the average diffusion layer thickness, xd, at the CE by means of the following relationship, 0.858 ( !zD~/G)“~ xd s, = 0.643 ,-+,1/‘5D1/3
(6)
=f
where u is the mean solution velocity (cm s-l) in the channel. We deduce a value of current flowing at the two kinds of electrode. These are given by [4],
f based on a comparison of the transport-limited IRDE LIM = 1.554 FAD2’3v-“6[Red]“w1/2 and
I&
= DFA[Red]“/a,
(7)
141 = 1.165 FD2’3(u/h)1’3
w12’3[Red]” = DFlw[Red]“/x,
(8)
where w is the width of the channel electrode and A the area of the rotating disc. Equations (6), (7) and (8) give a value for f of 1.00. That is to say that the RDE diffusion layer thickness may be simply interchanged with the average CE diffusion layer thickness. ‘Equation (6) together with our deduced value for f (of 1.00) allows the transformation of “working curves” deduced for one electrode into those relating to the other. The success of the transformation can be judged from the following examples. ECE mechanism This electrode reaction mechanism is defined (for a reduction) by the following scheme, A+e-+B B$C C+e-+G where C is more easily reduced than A. Such a process results in transport-limited currents which show a transition between two-electron and one-electron behaviour as the flow rate (a) or rotation speed (w) is increased. The theory describing the
291
1.6-
l-01*“* -0-S
n .I”.0
*I.‘* 0.5
*“*
* nr*“.’
1-o 1.5 ECE tog,,, K2,’ or WI, K,,
2.0
Fig. 2. Working curves for an ECE process at the rotating disc and channel electrodes.
effective number of electrons transferred, n,,, has been given for both the RDE [18] and the CE [14]. In the former case neff is dependent upon the parameter, Kg;
(9
= k(v/D)1’36’
and in the second case upon, K;zE = k( 12h2/9U2D)1’3
(10)
The relevant working curves are displayed in Fig. 2 [14,18]. Equations (a), (9) and (10) allow us to deduce that, log,, K:;’
= log,,K%,
+ log,, ([gy2)
= log,,K;$
- 0.568
9-l/3
(11)
(12) The x-axis of Fig. 2 relates to log,,KECE for the two electrode types. It can be seen that the two curves are, to a good approximation, parallel and separated by 0.568. Equation (12) is thus satisfactory and it may be concluded that eqn. (6) is valid in this particular example. DISPl
mechanism
This reaction mechanism is defined by A+e-+B B&Z B+Cfas’A+G
292
Fig. 3. Comparison of working curves for a DISPl process at the CE: (- - -) derived rigorously at the level of the LMque approximation [15]; ( . . . . . -) transformed via eqn. (6) from the corresponding curve for the RDE (181.
The scheme is closely related to the ECE mechanism and the relevant, normalised rate constants Kgyl and Kgispl are also given by eqns. (9) and (10) [15,18] except that k’ replaces k. We would thus expect, from the above, that DISPl KCE
o 1g-l/3~DI=‘l 0.643
= L
*
RDE
03)
The literature [15,18] allows the determination of nerf as a function of KDISP for both electrodes. Figure 3 shows neff p lotted against (the Napierian logarithm of) the two quantities, Kgispl and { [0.643/0.858]29-1/3K~~1}, in eqn. (13). The former curve is deduced from the CE theory [15]; the latter from the RDE theory [18]. The two curves are virtually indistinguishable, a result which again validates eqn. (6). EC’ (catabtic)
mechanism
This reaction mechanism is defined by A+e-+B B + Z %A
+ products
The products of the homogeneous step are taken to be electro-inactive at the potential of interest and the concentration of B is taken to be much less than that of Z so that the chemical step can be assumed as first-order. The effective number of electrons transferred under transport-limited conditions, has, in the case of the CE,
293
Fig. 4. Working curve for an EC’ mechanism at the CE: (- - -) derived rigorously by Matsuda et al. 1161;(. . . . . . ) calculated from eqn. (6) from the corresponding behaviour at the RDE 1191.
been shown by Matsuda parameter, The corresponding co-workers [ 191 K ;;;=
and co-workers
parameter
[16] to be related
for the RDE has been given by Northing
(k”/(8.032)2’3)W-l(~,/@“3
who employed numerical methods to generate the associated From eqns. (6), (14) and (15) we see that KEC, _ 0.643 x (8.032)“3 CE
to the following
-
(K;&)l’*
“working
and
(15) curve”.
(16)
0.858 x 3r13 Figure 4 shows n eff plotted against K,,Ec’ from the authentic CE calculation and also the right hand side of eqn. (16) (the CE parameter “deduced” from the RDE calculation via eqn. 6). The data come from refs. 16 and 19. A perfect match is found. CE mechanism
This mechanism is defined as follows, A&B
K=jf/i
ii
B + e- -+products
294
Fig. 5. neff -[log,&$ (- - -) or trode for a CE mechanism characterised by
1.22[ ~g&]~‘*} =l.
(.
. . . . .)] behaviour at a channel elec-
In the following we consider the particular case of K = 1, for which the relevant normalised rate constants are known [17,19] to be KCCE” = ( h1/3U)1’3(2z)“20’/6
07)
and Kz&=
(k/(8.032)2’3)w-1(v/0)1’3
08)
Application of eqn. (6) to these definitions gives
Kzf= 1.22 (K&)l'*
(19)
which enables us to compare the “working curves” for the conventional CE calculation and that generated from the RDE calculation. Figure 5 shows neff plotted against log,,K~~
and log,,(1.22(
Kg,)l'*).The data have been obtained
from refs. 17 and 19. Good agreement is apparent.
295
Potential step chronoamperometry We consider next the chronoamperometric response to a potential step from a region in which no current flows to a potential corresponding, under steady-state conditions, to the passage of the transport-limited current. This response has been calculated for the RDE by Mason and co-workers [19] who defined the following normalised time variable, rRDE= (8.032)2’3w(D/v)“3t
Fig. 6. Chronoamperometric response of a CE to a potential step such that reversible electron transfer becomes mass-transport limited. Rigorous theory (- - -) derived by Aoki et al. [20] is compared with the behaviour predicted by transforming the corresponding RDE curve [19] to the channel electrode via eqn. (6) (. . . . . .).
296
where t is real time. The equivalent quantity at the CE has been given by Aoki et al.
Px rcE= (3U/h1)2’3D”3t
(21)
Equations (20) and (21) may again be related via eqn. (6), .rcE = l.O84r,,,
(22) Figure 6 shows the chronoamperomogram calculated by Aoki et al. [20] for a CE and that derived for the same geometry through transformation of Mason and co-workers’ RDE data [19] through application of eqn. (22). Reasonable agreement is seen.
Current-potential curves As a final example we consider the general shape of the current-potential for the following simple electrode reaction, Ox+ne-
curves
SRed
k’,
where exp[-&]
(23)
k!., = k; exp[(l - a),$]
(24)
k;=kA and
.$=nF/RT(E-E”)
(25)
(Y is the transfer coefficient. expression electrodes;
I=
describes
It has been demonstrated [21-231 that the following the current-potential curve at a variety of hydrodynamic
I LIM ,C I I LIM,A 1 + e[
1 + e-<
(kA/D2/3)u[e-a’E A + (kA/D2’3).[e-crt
+ e(l-m)E] + e(l-a)E]
(26)
where A and u are electrode dependent. I,,,,, is the limiting current in the positive direction and is given by eqn. (7) or (8). IriM,c is the same current in the cathodic direction and may be calculated from the same equations except that [Ox] replaces [Red]. In the particular case of the CE [23], ccn = o- i/sjri/31i/3
(27)
and A,,
= 1.117
(28) Then combining eqns. (26), (27), (28) and (6) allows us to deduce that for the RDE, 0 RDE- - r,i/6(2,rw)-1/2
(29)
A mE = 0.594
(30)
291
Equation (29) is identical to that reported in the literature [21,22] and eqn. (30) compares extremely well with the reported value of 0.620. Again the transformation is shown to be remarkably successful. CONCLUSIONS AND CAVEAT
The level of agreement displayed between rigorous theory and that generated by the application of the transformation in eqn. (6) is highly satisfactory. We are therefore led to encourage the use of this transformation in the generation of working curves, or equations, for various problems at either the CE or the RDE when the corresponding problem has been solved for the other electrode type. We would however stress that quantitative success cannot be guaranteed a priori, although we know of no case where the transformation fails. We would anticipate in particular that problems involving second- (or higher) order kinetics are not likely to be as successfully predicted in this way. The physical reason for this is that in the case of a simple electron transfer (E process), although the limiting-current density depends on xp113, over most of the electrode length (apart from the upstream edge) the current density is not far removed from its average value. This is the reason for the success of the transformation outlined above and also the basis for the validity of the Singh and Dutt approximation (see above). The argument continues to hold in the case of first-order kinetics, but the effect of second- or higher order kinetics is to exaggerate the contribution of the upstream part of the electrode resulting in the collapse of the transformation. ACKNOWLEDGEMENT
We thank Ciba-Geigy for P.R.U.
Industrial Chemicals and SERC
for a CASE studentship
REFERENCES 1 R.G. Compton and P.R. Unwin, J. EIectroanaI. Chem., 205 (1986) 1. 2 R.G. Compton and P.R. Unwin in R.G. Compton (Ed.), Comprehensive Chemical Kinetics, Vol. 29, Elsevier, Amsterdam, in press. 3 W.G. Co&ran, Proc. Cambridge Philos. Sot., Math. Phys. Sci., 30 (1934) 345. 4 V.G. Levich, Physicochemical Hydrodynamics, Prentice-Hall, EngIewood Cliffs, NJ, 1962. 5 J.B. Flanagan and L. Marcoux, J. Phys. C&em., 78 (1974) 718. 6 K. Aoki, K. Tokuda and H. Matsuda,J~ElectroanaI. Chem., 217 (1987) 33. 7 M.A. L&que, Ann. Mines, Mem., Ser. 12, 13 (1928) 201. 8 J. Dutt and T. Singh, J. EIectroanal. Chem., 182 (1985) 259. 9 R.G. Compton, M.B.G. Pilkington, G.M. Steam and P.R. Unwin, J. Electroanal. Chem., 238 (1987) 43. 10 T. Singb and J. Dutt, J. Electroanal Chem., 190 (1985) 65. 11 T. Singh and J. Dutt, J. Electroanal. Chem., 196 (1985) 35. 12 J. Dutt and T. Singh, J. Electroanal Chem., 207 (1986) 11. 13 V. Yu. Fihnovski and Yu. V. Pleskov in E. Yeager, J.O’M. Bockris, B.E. Conway and S. Sarangapani (I?&.), Comprehensive Treatise of Electrochemistry, Vol. 9, Plenum Press, New York, 1984, p. 298.
298 14 15 16 17 18 19 20 21
R.G. Compton, D.J. Page and G.R. Sealy, J. Electroanal. Chem., 161 (1984) 129. R.G. Compton, P.J. Daly, P.R Unwin and A.M. Waller, J. Electroanal. Chem., 191 (1985) 15. K. Aoki, K. Tokuda and H. Matsuda, J. Electroanal. Chem., 76 (1977) 217. K. Tokuda, K. Aoki and H. Matsuda, J. Electroanrd. Chem., 80 (1977) 211. L.S. Marcoux, RN. Adams and S.W. Feldberg, J. Phys. Chem., 73 (1969) 2611. R.G. Compton, M.E. Laing, D. Mason, R.J. Northing and P.R. Unwin, Proc. R. Sot., in press. K. Aoki, K. Tokuda and H. Matsuda, J. Electroanal. Chem., 209 (1986) 247. C.M.A. Brett and A.M.C.F. Oliveira Brett in C.H. Bamford and R.G. Compton (Eds.), Comprehensive Chemical Kinetics, Vol. 26, Elsevier, Amsterdam, 1986, p. 404. 22 J. Koutecky and V.G. Levi& Dokl. Akad. Nauk SSR, 117 (1957) 422. 23 H. Matsuda, J. Electroanal. Chem., 35 (1972) 77.
J. Electroanal. Chem., 245 (1988) 287-298 Elsevier Sequoia S.A., Lausanne - Printed
in The Netherlands
THE RELATIONSHIP BETWEEN MASS TRANSPORT AND ROTATING DISK ELECTRODES
PATRICK
R. UNWIN
and RICHARD
Physical Chemistry Laboratory, (Received
TO CHANNEL
G. COMPTON
South Parks Road, Oxford OXI 3Q.Z (Great Britain)
29th June 1987; in revised form 29th September
1987)
ABSTRACT A simple equation is presented which relates the diffusion layer thickness at a rotating disk electrode to the average diffusion layer thickness at a channel (or tubular) electrode, under the assumption that the hydrodynamic flow in both cases is laminar. It is demonstrated that, through the equation, the parameters characterising various possible electrode reaction mechanisms (ECE, DISPl, EC’, CE,.. .) under transport-limited conditions at one of the two electrodes, are readily deduced once the corresponding mass transport problem has been rigorously solved at the other electrode. In addition, the relationship is shown to be equally successful when considering the chronoamperometric response at the two electrodes to a potential step, from a region in which no current flows to one corresponding, under steady-state conditions, to the passage of the transport-limited current, and also for the description of the general form of the current-potential curves for the two electrode geometries. The circumstances under which the transformation may be applied are critically assessed.
INTRODUCTION
The current intense activity in the characterisation of electrode reaction mechanisms has led to the theoretical description of a wide range of mechanisms at both rotating disc (RDE) and channel (CE) electrodes. The former constitutes probably the most commonly employed type of (solid) hydrodynamic electrode, whereas the advantages of the latter have been described recently [1,2]. Superficially the problem of mass transport to the two electrodes appears quite different, as can be seen by considering the relevant convective-diffusion equations for both cases. For the transport of a kinetically stable species of concentration, c, to a RDE we have [3],
ac a% ;i;=Dax’-““ax
ac
u, c - ~.O~~W~‘*Y-“~X~(Xe 0)
(1)
where x is the distance normal to, and starting from the surface of the electrode (see 0022-0728/88/$03.50
0 1988 Elsevier Sequoia
S.A.
288
(a)
2h
Fig. 1. Coordinate system for (a) the channel electrode and (b) the rotating disc electrode.
Fig. l), Y is the kinematic viscosity of the solution, o (Hz) is the disc rotation speed, and D is the diffusion coefficient of the species concerned. In contrast the corresponding CE equation is [4] a~ a2c ~=Dj--y-yjg
ac
ux=“o(l(y2)
(2)
where x is distance along the channel, y is the distance normal to the electrode surface and h is the half-height of the channel (Fig. 1). In writing eqn. (2) we have assumed that diffusion in directions other than normal to the electrode surface may be neglected. This approximation has been thoroughly explored and shown to be entirely satisfactory for practical electrodes [5,6]. In the treatment of CE problems it is usual to invoke the so-called Ltv&que approximation, originally introduced in respect of the corresponding heat transfer problem [7], u,-2u,y/h
(foryc0)
(3)
Comparison of the form at eqns. (1) and (3), which are valid for values of x close to the electrode surface, shows that the velocity profile is parabolic in x at the RDE, but linear at the CE. A further comparison of eqns. (1) and (2) indicates that the concentration profiles at the RDE resulting from the solution of the convective-diffusion equation will depend on x only - the RDE being “uniformly accessible” - but those at the CE will depend on both x and y. However, a further simplification in respect of
289
eqn. (2) has very recently been adopted by Dutt and Sir@ [8]. This involves replacing the concentration gradient in the x-direction by “its average value” [8]. Mathematically this means that for the one-electron electrode reaction, Red%Ox+ewe can write a[Red]/ax
= ([Red] - [Red]“)/l
(4)
and a[ox]/ax
= [ox]/z
(5)
where I is the electrode length and [Red] O is the bulk concentration of the electroactive species ([Ox]” is assumed to be zero). These approximations reduce eqn. (2) to an ordinary differential equation (in JJ). It has been shown previously [9] that theoretical results generated in this way are remarkably successful - when compared with full rigorous theory - for a wide range of problems including both time-dependent phenomena (such as chronoamperometry [9], linear sweep and cyclic voltammetry [8,10-121) and also complex steady-state problems (such as the current-flow rate behaviour of ECE and DISPl processes [9]). In other words, the question of mass transport to the CE can be treated satisfactorily as a one-dimensional problem within an “average” diffusion layer generated by the Singh and Dutt approximation - at least where the problem involves either stable species or species decaying by first-order kinetics [9]. Historically the RDE was the first solid hydrodynamic electrode to be widely adopted (see eg. ref. 13). Only relatively recently have the advantages of the CE been appreciated [1,2] - early workers probably being deterred because of the then wrongly perceived difficulties of handling mass transport problems at non-uniformly accessible electrodes [1,2]. Consequently, the literature describing the parameters (effective number of electrons transferred under limiting current conditions, half-wave potential,. . . ) which characterise the various possible mechanisms (ECE, DISP, EC’, CE, . . . ) o f an electrode reaction is far richer for the RDE than for the CE, although some theoretical treatments for the latter electrode have recently become available [14-171 - generally built on the LCv&que approximation (see above). Such parameters are normally reported in the form of “working curves” in which the dependence of the parameter of interest on an appropriate normalised variable (which depends typically on a rate constant, w or ua and D) is illustrated graphically and this diagram is then used for the analysis of experimental data. Since the publication of various “working curves” for the CE we have been impressed with the apparent similarity in shape of the corresponding curves for the two electrodes, although this is perhaps not unexpected in the light of the Singh and Dutt Approximation. Consequently, we have investigated whether a simple transformation exists by which the behaviour of a particular process at one electrode type may be deduced once the corresponding problem has been solved at the other electrode. This paper suggests the appropriate transformation in terms of a
290
simple direct relationship between the diffusion layer thickness at the two electrodes - and demonstrates the successful application of the transformation in a diversity of problems. The conditions under which the transformation may be applied are carefully assessed. THE TRANSFORMATION
AND ITS APPLICATIONS
We suggest that, to a reasonable approximation, the diffusion layer thickness, S,, at the RDE can be related to the average diffusion layer thickness, xd, at the CE by means of the following relationship, 0.858 ( !zD~/G)“~ xd s, = 0.643 ,-+,1/‘5D1/3
(6)
=f
where u is the mean solution velocity (cm s-l) in the channel. We deduce a value of current flowing at the two kinds of electrode. These are given by [4],
f based on a comparison of the transport-limited IRDE LIM = 1.554 FAD2’3v-“6[Red]“w1/2 and
I&
= DFA[Red]“/a,
(7)
141 = 1.165 FD2’3(u/h)1’3
w12’3[Red]” = DFlw[Red]“/x,
(8)
where w is the width of the channel electrode and A the area of the rotating disc. Equations (6), (7) and (8) give a value for f of 1.00. That is to say that the RDE diffusion layer thickness may be simply interchanged with the average CE diffusion layer thickness. ‘Equation (6) together with our deduced value for f (of 1.00) allows the transformation of “working curves” deduced for one electrode into those relating to the other. The success of the transformation can be judged from the following examples. ECE mechanism This electrode reaction mechanism is defined (for a reduction) by the following scheme, A+e-+B B$C C+e-+G where C is more easily reduced than A. Such a process results in transport-limited currents which show a transition between two-electron and one-electron behaviour as the flow rate (a) or rotation speed (w) is increased. The theory describing the
291
1.6-
l-01*“* -0-S
n .I”.0
*I.‘* 0.5
*“*
* nr*“.’
1-o 1.5 ECE tog,,, K2,’ or WI, K,,
2.0
Fig. 2. Working curves for an ECE process at the rotating disc and channel electrodes.
effective number of electrons transferred, n,,, has been given for both the RDE [18] and the CE [14]. In the former case neff is dependent upon the parameter, Kg;
(9
= k(v/D)1’36’
and in the second case upon, K;zE = k( 12h2/9U2D)1’3
(10)
The relevant working curves are displayed in Fig. 2 [14,18]. Equations (a), (9) and (10) allow us to deduce that, log,, K:;’
= log,,K%,
+ log,, ([gy2)
= log,,K;$
- 0.568
9-l/3
(11)
(12) The x-axis of Fig. 2 relates to log,,KECE for the two electrode types. It can be seen that the two curves are, to a good approximation, parallel and separated by 0.568. Equation (12) is thus satisfactory and it may be concluded that eqn. (6) is valid in this particular example. DISPl
mechanism
This reaction mechanism is defined by A+e-+B B&Z B+Cfas’A+G
292
Fig. 3. Comparison of working curves for a DISPl process at the CE: (- - -) derived rigorously at the level of the LMque approximation [15]; ( . . . . . -) transformed via eqn. (6) from the corresponding curve for the RDE (181.
The scheme is closely related to the ECE mechanism and the relevant, normalised rate constants Kgyl and Kgispl are also given by eqns. (9) and (10) [15,18] except that k’ replaces k. We would thus expect, from the above, that DISPl KCE
o 1g-l/3~DI=‘l 0.643
= L
*
RDE
03)
The literature [15,18] allows the determination of nerf as a function of KDISP for both electrodes. Figure 3 shows neff p lotted against (the Napierian logarithm of) the two quantities, Kgispl and { [0.643/0.858]29-1/3K~~1}, in eqn. (13). The former curve is deduced from the CE theory [15]; the latter from the RDE theory [18]. The two curves are virtually indistinguishable, a result which again validates eqn. (6). EC’ (catabtic)
mechanism
This reaction mechanism is defined by A+e-+B B + Z %A
+ products
The products of the homogeneous step are taken to be electro-inactive at the potential of interest and the concentration of B is taken to be much less than that of Z so that the chemical step can be assumed as first-order. The effective number of electrons transferred under transport-limited conditions, has, in the case of the CE,
293
Fig. 4. Working curve for an EC’ mechanism at the CE: (- - -) derived rigorously by Matsuda et al. 1161;(. . . . . . ) calculated from eqn. (6) from the corresponding behaviour at the RDE 1191.
been shown by Matsuda parameter, The corresponding co-workers [ 191 K ;;;=
and co-workers
parameter
[16] to be related
for the RDE has been given by Northing
(k”/(8.032)2’3)W-l(~,/@“3
who employed numerical methods to generate the associated From eqns. (6), (14) and (15) we see that KEC, _ 0.643 x (8.032)“3 CE
to the following
-
(K;&)l’*
“working
and
(15) curve”.
(16)
0.858 x 3r13 Figure 4 shows n eff plotted against K,,Ec’ from the authentic CE calculation and also the right hand side of eqn. (16) (the CE parameter “deduced” from the RDE calculation via eqn. 6). The data come from refs. 16 and 19. A perfect match is found. CE mechanism
This mechanism is defined as follows, A&B
K=jf/i
ii
B + e- -+products
294
Fig. 5. neff -[log,&$ (- - -) or trode for a CE mechanism characterised by
1.22[ ~g&]~‘*} =l.
(.
. . . . .)] behaviour at a channel elec-
In the following we consider the particular case of K = 1, for which the relevant normalised rate constants are known [17,19] to be KCCE” = ( h1/3U)1’3(2z)“20’/6
07)
and Kz&=
(k/(8.032)2’3)w-1(v/0)1’3
08)
Application of eqn. (6) to these definitions gives
Kzf= 1.22 (K&)l'*
(19)
which enables us to compare the “working curves” for the conventional CE calculation and that generated from the RDE calculation. Figure 5 shows neff plotted against log,,K~~
and log,,(1.22(
Kg,)l'*).The data have been obtained
from refs. 17 and 19. Good agreement is apparent.
295
Potential step chronoamperometry We consider next the chronoamperometric response to a potential step from a region in which no current flows to a potential corresponding, under steady-state conditions, to the passage of the transport-limited current. This response has been calculated for the RDE by Mason and co-workers [19] who defined the following normalised time variable, rRDE= (8.032)2’3w(D/v)“3t
Fig. 6. Chronoamperometric response of a CE to a potential step such that reversible electron transfer becomes mass-transport limited. Rigorous theory (- - -) derived by Aoki et al. [20] is compared with the behaviour predicted by transforming the corresponding RDE curve [19] to the channel electrode via eqn. (6) (. . . . . .).
296
where t is real time. The equivalent quantity at the CE has been given by Aoki et al.
Px rcE= (3U/h1)2’3D”3t
(21)
Equations (20) and (21) may again be related via eqn. (6), .rcE = l.O84r,,,
(22) Figure 6 shows the chronoamperomogram calculated by Aoki et al. [20] for a CE and that derived for the same geometry through transformation of Mason and co-workers’ RDE data [19] through application of eqn. (22). Reasonable agreement is seen.
Current-potential curves As a final example we consider the general shape of the current-potential for the following simple electrode reaction, Ox+ne-
curves
SRed
k’,
where exp[-&]
(23)
k!., = k; exp[(l - a),$]
(24)
k;=kA and
.$=nF/RT(E-E”)
(25)
(Y is the transfer coefficient. expression electrodes;
I=
describes
It has been demonstrated [21-231 that the following the current-potential curve at a variety of hydrodynamic
I LIM ,C I I LIM,A 1 + e[
1 + e-<
(kA/D2/3)u[e-a’E A + (kA/D2’3).[e-crt
+ e(l-m)E] + e(l-a)E]
(26)
where A and u are electrode dependent. I,,,,, is the limiting current in the positive direction and is given by eqn. (7) or (8). IriM,c is the same current in the cathodic direction and may be calculated from the same equations except that [Ox] replaces [Red]. In the particular case of the CE [23], ccn = o- i/sjri/31i/3
(27)
and A,,
= 1.117
(28) Then combining eqns. (26), (27), (28) and (6) allows us to deduce that for the RDE, 0 RDE- - r,i/6(2,rw)-1/2
(29)
A mE = 0.594
(30)
291
Equation (29) is identical to that reported in the literature [21,22] and eqn. (30) compares extremely well with the reported value of 0.620. Again the transformation is shown to be remarkably successful. CONCLUSIONS AND CAVEAT
The level of agreement displayed between rigorous theory and that generated by the application of the transformation in eqn. (6) is highly satisfactory. We are therefore led to encourage the use of this transformation in the generation of working curves, or equations, for various problems at either the CE or the RDE when the corresponding problem has been solved for the other electrode type. We would however stress that quantitative success cannot be guaranteed a priori, although we know of no case where the transformation fails. We would anticipate in particular that problems involving second- (or higher) order kinetics are not likely to be as successfully predicted in this way. The physical reason for this is that in the case of a simple electron transfer (E process), although the limiting-current density depends on xp113, over most of the electrode length (apart from the upstream edge) the current density is not far removed from its average value. This is the reason for the success of the transformation outlined above and also the basis for the validity of the Singh and Dutt approximation (see above). The argument continues to hold in the case of first-order kinetics, but the effect of second- or higher order kinetics is to exaggerate the contribution of the upstream part of the electrode resulting in the collapse of the transformation. ACKNOWLEDGEMENT
We thank Ciba-Geigy for P.R.U.
Industrial Chemicals and SERC
for a CASE studentship
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R.G. Compton, D.J. Page and G.R. Sealy, J. Electroanal. Chem., 161 (1984) 129. R.G. Compton, P.J. Daly, P.R Unwin and A.M. Waller, J. Electroanal. Chem., 191 (1985) 15. K. Aoki, K. Tokuda and H. Matsuda, J. Electroanal. Chem., 76 (1977) 217. K. Tokuda, K. Aoki and H. Matsuda, J. Electroanrd. Chem., 80 (1977) 211. L.S. Marcoux, RN. Adams and S.W. Feldberg, J. Phys. Chem., 73 (1969) 2611. R.G. Compton, M.E. Laing, D. Mason, R.J. Northing and P.R. Unwin, Proc. R. Sot., in press. K. Aoki, K. Tokuda and H. Matsuda, J. Electroanal. Chem., 209 (1986) 247. C.M.A. Brett and A.M.C.F. Oliveira Brett in C.H. Bamford and R.G. Compton (Eds.), Comprehensive Chemical Kinetics, Vol. 26, Elsevier, Amsterdam, 1986, p. 404. 22 J. Koutecky and V.G. Levi& Dokl. Akad. Nauk SSR, 117 (1957) 422. 23 H. Matsuda, J. Electroanal. Chem., 35 (1972) 77.